The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

Highlights

• A non-scaling method is utilized to derive conditions for the quasi-steady-state approximations.

• A parameter determining the validity of the reverse quasi-steady-state approximation is derived.

• A dynamical transcritical bifurcation is shown to occur in the phase-plane of the model.

• Our analysis provides a deeper understanding of the quasi-steady-state approximation.

Abstract

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov–Fenichel parameters and their respective critical manifolds. By combining Tikhonov–Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new “small” parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

Introduction

Perhaps the most well-known reaction in biochemistry is the Michaelis–Menten (MM) reaction mechanism, (1), which describes the catalytic conversation of a substrate, $S$, into a product, $P$. Catalysis is achieved through means of an enzyme, $E$, that reversibly binds with the substrate, forming an intermediate complex, $C$. In turn, $C$ irreversibly disassociates into enzyme and product molecules:

$$S+E\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons }}C\stackrel{k_2}{⟶}E+P.$$

For theorists and applied mathematicians, whose aim is to mathematically describe the dynamics of chemical networks and metabolic pathways [1], [2], the MM reaction mechanism serves as a building block to these more complex systems. In general, there are two ways to mathematically model enzyme catalyzed reactions. At low concentrations of chemical species stochastic models are generally favorable since they describe the seemingly random collisions of reactant molecules in intracellular environments [3], [4]. In contrast, when concentrations are high and the chemical species are well-mixed, the MM mechanism can be appropriately modeled as a set of nonlinear ordinary differential equations. Although one can argue that deterministic models are in some sense more manageable than stochastic models, nonlinear deterministic models rarely admit closed form solutions, and therefore model reduction techniques must be employed in order to simplify the model equations so that approximate solutions can be obtained and analyzed. Typically, model reduction is synonymous with approximating the model dynamics on an invariant manifold, and the advent of powerful computer algebra systems aids in the systematic reduction of high-dimensional or even infinite-dimensional (i.e., partial differential equations and delay differential equations) dynamical systems. Not surprisingly, there is a large body of literature that clearly illustrates how model reduction can uncover, quantify, and explain the various nonlinear phenomena arising not only in chemistry, but also in biology [5], [6], [7] and physics [8], [9], [10], [11].

Historically, the most widely-utilized reduction technique in deterministic enzyme kinetics has been the singular perturbation method [12], in which a reduced model is constructed by approximating the flow of the model equations on a slow invariant manifold (SIM). Since the dimension of the SIM is less than the dimension of the phase-space, approximation of the dynamics on the SIM permits a reduction in the dimension of the problem. The singular perturbation method exploits the presence of disparate fast and slow timescales, and the work of Tikhonov [13] and Gradshtein [14] provides the necessary rigorous foundation for the construction of a reduced model. Briefly, when fast and slow timescales are present, the differential equations that model the MM reaction with a timescale separation can be expressed in the form

$$\dot{x}=f(x,y;\epsilon ),$$$$\epsilon \dot{y}=g(x,y;\epsilon ),$$

where $0<\epsilon \ll 1$. Setting $\epsilon =0$ yields a differential–algebraic-equation

$$\dot{x}=f(x,y;0),$$$$0=g(x,y;0),$$

and, according to Tikhonov’s theorem, Eqs. (3a)–(3b) provide a very good approximation to the dynamics of (2a)–(2b) when $\epsilon$ is sufficiently small. The reduced system (3a)–(3b) provides a simpler, and often times more tractable, reduced mathematical model that is commonly referred to as a quasi-steady-state approximation (QSSA).

The hope is that the condition that supports the validity of (3a)–(3b) (i.e., $\epsilon \ll 1$) is easy to implement in enzymatic assays, so that precise and accurate measurements of the kinetic parameters pertinent to the reaction can be made by fitting the QSSA model to experimental time course data [15], [16]. Due to the necessity of disparate timescales, slow manifold reduction may not be possible in every physical scenario. Therefore, the challenge for theorists is not only to derive a reduced model that has suitable utility, but also to determine the unique physical and chemical conditions that permit the validity of the associated reduction. Thus, an important task of the theoretician is determine the exact conditions for which the reduced model is valid [17]. Mathematically, this translates to determining “$\epsilon$”, a (typically) dimensionless parameter that may not be unique. The non-uniqueness of “$\epsilon$” adds complication, since some “epsilons” are better than others. The best-known example of the “non-uniqueness dilemma” resides the history of the derivation of the Michaelis–Menten (MM) equation

$$v=\frac{Vs}{K_M+s},$$

which is obtained by applying the QSSA to the MM reaction mechanism (1). In (4), $v$ is the velocity of product formation in the reaction, $V$ is the limiting rate of the reaction, $K_M$ is the Michaelis constant, and $s$ is the free substrate concentration for the reaction. Alternatively, the MM equation is often referred to as the standard quasi-steady-state approximation (sQSSA). In 1967, Heineken et al. [18] formally applied, for the first time, the standard QSSA to the nonlinear differential equations governing the MM reaction mechanism (1) via singular perturbation analysis. Based on the findings of Laidler [19], a pioneer in chemical kinetics and authority on the physical chemistry of enzymes, Heineken et al. [18] introduced a consistent “$\epsilon$” for the MM reaction mechanism through scaling analysis. A more rigorous analysis of the sQSSA was introduced by Reich and Sel’kov [20] and Schauer and Heinrich [21], from which other “epsilons” where determined by proposing conditions that minimized the errors in the implementation of the sQSSA. In 1988, Lee A. Segel [22] derived the widely accepted criterion for the validity of the sQSSA and the derivation of the MM equation (4) by estimating the slow and fast timescales of the reaction. Segel [22] illustrated that prior physico-chemical knowledge about the reaction dynamic is instrumental in deriving the fast and slow timescales and uncovering the most general criteria for the validity of the sQSSA. As a direct result of Segel’s scaling method, the conditions for the validity of the sQSSA were derived for suicide substrates [23], [24], alternative substrates [25], fully competitive enzyme reactions [26], zymogen activation [27], and coupled enzyme assays [28], [29]. Segel’s scaling approach was also applied to the analysis of the MM reaction mechanism (1) to extend the validity of the QSSA in different regions of the initial enzyme and substrate concentration parameter space via the reverse QSSA (rQSSA) [30], [31], and the total QSSA (tQSSA) [32], [33], [34].

Despite the power of Segel’s scaling and simplification analysis for the MM reaction mechanism (1), there is still a fundamental challenge with its implementation via the rQSSA: there has never been a small parameter (i.e., a specific “$\epsilon$”), analogous to the one obtained by Segel [22] for the sQSSA, that is as effective at determining when the rQSSA is valid. Unfortunately, estimating the fast timescale associated with the rQSSA is difficult, and there have been fundamental disagreements in the reported estimates [31]. Thus, timescale estimation continues to be the “Achilles’ heel” of the rQSSA. This raises the obvious question of whether or not timescale estimation is truly the best approach towards resolving this problem.

Recently, Walcher and his collaborators [35], [36], [37] demonstrated that identifying a Tikhonov–Fenichel parameter (TFP) is an effective way to determine a priori conditions that suggest the validity of the QSSA. Essentially, a TFP is a dimensional parameter – such as a rate constant or initial concentration of a species – that, when identically zero, ensures the existence of a manifold of equilibrium points. Such manifolds are central to geometric singular perturbation theory (GSPT) and, as a result of Fenichel [38], it is well-understood that when certain conditions hold, the existence of a critical manifold of equilibria ensures the existence of a SIM once the TFP is small but non-zero. In this sense the origin of the SIM can be linked to the vanishing of a specific dimensional parameter, and a sufficiently small TFP ensures the existence of a SIM and the corresponding validity of a QSSA. However, the identification of a TFP does not diminish the importance of the asymptotic small parameter $\epsilon$, since it is essential to define what physically constitutes “small” when a parameter is non-zero. In other words, we must still answer the question: how small should Tikhonov–Fenichel parameters be in comparison to other dimensional parameters in order to yield an accurate reduced model?

In this paper, our primary objective is to determine a specific small parameter that determines the validity of the rQSSA to the MM reaction mechanism (1), but also to convey subject matter that can be quite technical in a language capable of reaching a wider audience that is not limited to applied mathematicians and physical chemists. In Section 2, we review the conditions for the validity of the various quasi-steady-state approximations that are commonly employed to approximate the long-time dynamics of the MM reaction mechanism (1): namely, the sQSSA, the rQSSA, and tQSSA. In Section 3, we assess the validity of specific QSS reductions by employing geometric singular perturbation theory, and illustrate how assumptions about the validity of the sQSSA based on Segel’s timescale separation can lead to erroneous conclusions. In Section 4, we introduce two methods that do not rely on scaling or timescale separation: Tikhonov–Fenichel parameters and energy methods, both of which can be employed to determine the validity of the QSSA. In Section 5, we analyze the validity of the rQSSA using the methods discussed in Section 4, and in Section 6, we discuss timescale separation and the hierarchy of small parameters that support the justification of the sQSSA, rQSSA, and the tQSSA. Finally, in our discussion (Section 7), we summarize our results and critique some of the conclusions drawn about the validity of the sQSSA and the rQSSA in the previous analyses of Segel and Slemrod [30], and Schnell and Maini [31], respectively. We also discuss the impact our results will have on experimental assays, and how the methods we utilize can be employed to analyze more complicated reactions and experimental assays.